Mathematics Weblog

Coin tossing is not fair

Saturday 20 March 2004 at 11:02 pm | In Articles | Post Comment

If you toss a coin and want it to come up heads then start with it heads up. 51% of the time it will come up heads. This remarkable result was discovered by a mathematician Persi Diaconis and you can hear an interview with him here (then click on All Things Considered audio link) where he talks about the lack of randomness of coin tossing.

It turns out that any randomness is just due to sloppy humans who toss coins differently every time. So all those text books on tossing coins are wrong and until now nobody had noticed!

Division by zero shock!

Tuesday 16 March 2004 at 4:35 pm | In Articles | Post Comment

It is well known that division by zero is the ultimate mistake to make in mathematics (see Dividing by 0 and Wikipedia). Different rules may apply in other advanced number systems but for use in the set of real numbers it has to be avoided. Doing so is often not easy, but one has to live with the consequences.

It is thus a shock to see division by zero hidden in a question in the AQA A Level Mathematics Specimen P4 Paper. Question 7 starts with

$\begin{displaymath}\text{f} : x\mapsto\tan3x-4x \end {displaymath}$

$0\le x \le \frac{\pi}{6}$

Where’s the division by zero? It is implicit in the definition of $\tan\theta$ which is defined as $\dfrac{\sin\theta}{\cos\theta}$ , provided of course that $\cos\theta\not=0$ ; so what happens in the given domain? Thus doesn’t exist, which makes the rest of the question unanswerable, since it requires you to differentiate and integrate 😀

The examiners should have known better. Perhaps it is just as well that the syllabus (and hence the specimen paper) will be replaced, starting in 2004.

Thanks to one of my eagle-eyed students who spotted this horror!

Pi

Sunday 14 March 2004 at 6:35 pm | In Articles | Post Comment

The digits in the decimal of $\pi$ have always intrigued mathematicans. In the computer age $\pi$ has been calculated to billion of places. You can see the first 30 million digits at The Inverse Symbolic Calculator.

$\pi$ is known to be irrational (so the decimal expansion is infinite and non-recurring) and transcendental (so it is not a root of a polynomial equation with rational coefficients). You can see a proof on my page here. Yet it isn’t known if the digits are random in the statistical sense.

One of the things that doesn’t help in examining the decimal digits of $\pi$ is that if you want to know, say, the 1000^th decimal digit then you will have to calculate the previous 999.

Note the word decimal in the last paragraph. If you go over to hex digits (base 16) then you can calculate any digit without knowing the previous ones. This truly remarkable result was discovered in 1996 by David Bailey, Peter Borwein and Simon Plouffe (their paper can be found here). What is astonishing is that the formula for $\pi$ used to calcuate the n^th is simple enough to be shown here:

$\displaystyle \pi = \sum _{k=0} ^{\infty} \dfrac{1}{16^k} \left[\dfrac{4}{8k+1}-\dfrac{2}{8k+4}-\dfrac{1}{8k+5}-\dfrac{1}{8k+6}\right]$